An Asymptotic Preserving and Energy Stable Scheme for the Euler-Poisson System in the Quasineutral Limit

TL;DR

This paper introduces an asymptotic preserving and energy stable numerical scheme for the Euler-Poisson system in the quasineutral limit, ensuring physical consistency and robustness across different regimes.

Contribution

It develops a novel semi-implicit finite volume scheme with correction terms that guarantee energy stability, entropy dissipation, and asymptotic preservation in the quasineutral limit.

Findings

The scheme is entropy stable and energy dissipative.

It remains accurate and stable as the Debye length tends to zero.

Numerical tests confirm robustness and efficiency.

Abstract

An asymptotic preserving and energy stable scheme for the Euler-Poisson system under the quasineutral scaling is designed and analysed. Correction terms are introduced in the convective fluxes and the electrostatic potential, which lead to the dissipation of mechanical energy and the entropy stability. The resolution of the semi-implicit in time finite volume in space fully-discrete scheme involves two steps: the solution of an elliptic problem for the potential and an explicit evaluation for the density and velocity. The proposed scheme possesses several physically relevant attributes, such as the the entropy stability and the consistency with the weak formulation of the continuous Euler-Poisson system. The AP property of the scheme, i.e. the boundedness of the mesh parameters with respect to the Debye length and its consistency with the quasineutral limit system, is shown. The results of numerical case studies are presented to substantiate the robustness and efficiency of the proposed method.